Spatial Point Patterns Analysis

Author

Jiale SO

Published

August 30, 2024

Modified

September 2, 2024

1.0 Context:


Spatial Point Pattern Analysis involves evaluating the pattern or distribution of a set of points on a surface. These points can represent the locations of:

  • Events, such as crimes, traffic accidents, or disease outbreaks, or

  • Business services, like coffee shops and fast food outlets, or facilities such as childcare and eldercare centers.

The specific questions we would like to answer are as follows:

  • are the childcare centres in Singapore randomly distributed throughout the country?

  • if the answer is not, then the next logical question is where are the locations with higher concentration of childcare centres?

2.0 Downloading the Data setsss


To provide answers to the questions above, three data sets will be used. They are:

  • CHILDCARE, a point feature data providing both location and attribute information of childcare centres. It was downloaded from Data.gov.sg and is in geojson format. Link here

  • MP14_SUBZONE_WEB_PL, a polygon feature data providing information of URA 2014 Master Plan Planning Subzone boundary data. It is in ESRI shapefile format. This data set was also downloaded from Data.gov.sg. Link here

  • CostalOutline, a polygon feature data showing the national boundary of Singapore. It is provided by SLA and is in ESRI shapefile format. Link here

3.0 Installing and loading R packages


In this hands-on exercise, five R packages will be used, they are:

  • sf, a relatively new R package specially designed to import, manage and process vector-based geospatial data in R.

  • spatstat, which has a wide range of useful functions for point pattern analysis. In this hands-on exercise, it will be used to perform 1st- and 2nd-order spatial point patterns analysis and derive kernel density estimation (KDE) layer.

  • raster which reads, writes, manipulates, analyses and model of gridded spatial data (i.e. raster). In this hands-on exercise, it will be used to convert image output generate by spatstat into raster format.

  • maptools which provides a set of tools for manipulating geographic data. In this hands-on exercise, we mainly use it to convert Spatial objects into ppp format of spatstat.

  • tmap which provides functions for plotting cartographic quality static point patterns maps or interactive maps by using leaflet API.

Use the code chunk below to install and launch the five R packages.

pacman::p_load(sf, raster, spatstat, tmap, tidyverse, rvest, geojsonsf)

4.0 Spatial Data Wrangling

4.1 Importing Spatial Data

childcare_sf <- st_read("data/child-care-services-geojson.geojson") %>%
  st_transform(crs = 3414)
Reading layer `child-care-services-geojson' from data source 
  `C:\Users\jiale\Desktop\IS415\IS415-GAA\Hands_On_Exercises\Hands_On_Exercise_3\data\child-care-services-geojson.geojson' 
  using driver `GeoJSON'
Simple feature collection with 1545 features and 2 fields
Geometry type: POINT
Dimension:     XYZ
Bounding box:  xmin: 103.6824 ymin: 1.248403 xmax: 103.9897 ymax: 1.462134
z_range:       zmin: 0 zmax: 0
Geodetic CRS:  WGS 84
sg_sf <- st_read(dsn = "data", layer="CostalOutline")
Reading layer `CostalOutline' from data source 
  `C:\Users\jiale\Desktop\IS415\IS415-GAA\Hands_On_Exercises\Hands_On_Exercise_3\data' 
  using driver `ESRI Shapefile'
Simple feature collection with 60 features and 4 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 2663.926 ymin: 16357.98 xmax: 56047.79 ymax: 50244.03
Projected CRS: SVY21
mpsz_sf <- st_read(dsn = "data", layer = "MP14_SUBZONE_WEB_PL")
Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `C:\Users\jiale\Desktop\IS415\IS415-GAA\Hands_On_Exercises\Hands_On_Exercise_3\data' 
  using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21

4.1.1 DIY:Use the appropriate SF function to retrieve the referencing system information of these geospatial data.

Simple, use the st_crs function from SF to check and print the crs information

# Retrieve CRS information
childcare_crs <- st_crs(childcare_sf)
sg_crs <- st_crs(sg_sf)
mpsz_crs <- st_crs(mpsz_sf)

# Print CRS information
print(childcare_crs)
Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]
print(sg_crs)
Coordinate Reference System:
  User input: SVY21 
  wkt:
PROJCRS["SVY21",
    BASEGEOGCRS["SVY21[WGS84]",
        DATUM["World Geodetic System 1984",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]],
            ID["EPSG",6326]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["Degree",0.0174532925199433]]],
    CONVERSION["unnamed",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["(E)",east,
            ORDER[1],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]],
        AXIS["(N)",north,
            ORDER[2],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]]]
print(mpsz_crs)
Coordinate Reference System:
  User input: SVY21 
  wkt:
PROJCRS["SVY21",
    BASEGEOGCRS["SVY21[WGS84]",
        DATUM["World Geodetic System 1984",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]],
            ID["EPSG",6326]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["Degree",0.0174532925199433]]],
    CONVERSION["unnamed",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["(E)",east,
            ORDER[1],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]],
        AXIS["(N)",north,
            ORDER[2],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]]]

4.1.2 DIY: Assign the correct CRS to MPSZ_SF and SG_SF Simple Feature Data frames.

notice that the MPSZ_SF and SG_SF is in World Geodetic System 1984 format, we need set the correct crs to these data and we can do so using the st transform. We can do so using the transform method

mpsz_sf <- st_read(dsn = "data", layer = "MP14_SUBZONE_WEB_PL") %>%
  st_transform(crs = 3414)
Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `C:\Users\jiale\Desktop\IS415\IS415-GAA\Hands_On_Exercises\Hands_On_Exercise_3\data' 
  using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21
sg_sf <- st_read(dsn = "data", layer = "CostalOutline") %>%
  st_transform(crs = 3414)
Reading layer `CostalOutline' from data source 
  `C:\Users\jiale\Desktop\IS415\IS415-GAA\Hands_On_Exercises\Hands_On_Exercise_3\data' 
  using driver `ESRI Shapefile'
Simple feature collection with 60 features and 4 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 2663.926 ymin: 16357.98 xmax: 56047.79 ymax: 50244.03
Projected CRS: SVY21
print(st_crs(mpsz_sf))
Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]
print(st_crs(sg_sf))
Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

4.1.3 Change the referencing System to Singapore National Projected Coordinate System

Note

Understanding the CRS in Our Data:

  1. MPZ and Coastal Data:

    • CRS: SVY21, which is the Singapore National Projected Coordinate System based on WGS84.

    • Description: This is a common projected coordinate system used in Singapore for accurate mapping.

  2. Childcare Data:

    • CRS: SVY21 / Singapore TM (Transverse Mercator projection).

    • Description: This is also a projection based on SVY21, specifically using the Transverse Mercator projection. It is very close to the SVY21 system, with minor differences in how the projection is handled.

Given that the map file serves as the base, we want all spatial data to overlay correctly, we should:

  1. Transform the GeoJSON Data to Match the Map File’s CRS:

    • Since our MPZ and Coastal data are already in SVY21 (EPSG:3414), transform the GeoJSON data to EPSG:3414 as well.
  2. Rationale:

    • This approach ensures that the childcare locations from the GeoJSON data will be accurately plotted within the boundaries and context provided by the map file (MPZ and Coastal data).

    • It avoids potential issues with misalignment, especially since oour base map data is already set up in a local projection suitable for Singapore.

# Transform Childcare data to match the base map's CRS (EPSG:3414)
childcare_sf <- st_read("data/child-care-services-geojson.geojson") %>%
  st_transform(crs = 3414)
Reading layer `child-care-services-geojson' from data source 
  `C:\Users\jiale\Desktop\IS415\IS415-GAA\Hands_On_Exercises\Hands_On_Exercise_3\data\child-care-services-geojson.geojson' 
  using driver `GeoJSON'
Simple feature collection with 1545 features and 2 fields
Geometry type: POINT
Dimension:     XYZ
Bounding box:  xmin: 103.6824 ymin: 1.248403 xmax: 103.9897 ymax: 1.462134
z_range:       zmin: 0 zmax: 0
Geodetic CRS:  WGS 84
# Now, all datasets should be aligned in the same CRS

4.1.4 Checking for validity of maps

When working with spatial data, it’s crucial to ensure that all geometries are valid. Invalid geometries can cause errors in analysis and visualization.

  1. Checking Validity with st_is_valid():
  2. Identifying Invalid Geometries:
  3. Fixing Invalid Geometries with st_make_valid()
mpsz_validity <- st_is_valid(mpsz_sf)
mpsz_invalid <- which(!mpsz_validity)
if (length(mpsz_invalid) > 0) {
  print("MPZ Invalid!")
  print(mpsz_sf[mpsz_invalid, ])
} else {
  print("it's valid!")
}
[1] "MPZ Invalid!"
Simple feature collection with 9 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 12535.88 ymin: 21678.35 xmax: 56396.44 ymax: 49291.03
Projected CRS: SVY21 / Singapore TM
    OBJECTID SUBZONE_NO             SUBZONE_N SUBZONE_C CA_IND
19        19          2        SOUTHERN GROUP    SISZ02      N
20        20          1               SENTOSA    SISZ01      N
24        24          1       MARITIME SQUARE    BMSZ01      N
122      122          9           JURONG PORT    JESZ09      N
123      123          3               SAMULUN    BLSZ03      N
128      128          9                PANDAN    CLSZ09      N
258      258          4        PASIR RIS PARK    PRSZ04      N
302      302          1 NORTH-EASTERN ISLANDS    NESZ01      N
320      320          9           NORTH COAST    WDSZ09      N
               PLN_AREA_N PLN_AREA_C          REGION_N REGION_C
19       SOUTHERN ISLANDS         SI    CENTRAL REGION       CR
20       SOUTHERN ISLANDS         SI    CENTRAL REGION       CR
24            BUKIT MERAH         BM    CENTRAL REGION       CR
122           JURONG EAST         JE       WEST REGION       WR
123              BOON LAY         BL       WEST REGION       WR
128              CLEMENTI         CL       WEST REGION       WR
258             PASIR RIS         PR       EAST REGION       ER
302 NORTH-EASTERN ISLANDS         NE NORTH-EAST REGION      NER
320             WOODLANDS         WD      NORTH REGION       NR
             INC_CRC FMEL_UPD_D   X_ADDR   Y_ADDR SHAPE_Leng SHAPE_Area
19  5809FC547293EA2D 2014-12-05 29815.09 23412.59  25626.977    2206319
20  A6FCDC9C447952CB 2014-12-05 27593.94 25813.35  17496.194    4919132
24  C1AC31ABF9978DDB 2014-12-05 25805.79 27911.42  13737.116    2701634
122 0664CA7EF6504AE5 2014-12-05 15250.74 32183.92  11355.002    2464857
123 F78E0287D3F24214 2014-12-05 13418.49 32264.59   8738.679    1940693
128 A6EE4A49376B69C4 2014-12-05 19228.60 32265.40   5689.647    1312923
258 9856E3CDCF57AD96 2014-12-05 41529.80 40218.94   8533.964    1719705
302 92BC3E09C68F3B52 2014-12-05 50424.79 42612.88  62436.235   67250563
320 898B2436858382A1 2014-12-05 22147.04 48031.55  10847.882    2450784
                          geometry
19  MULTIPOLYGON (((29712.51 23...
20  MULTIPOLYGON (((26858.1 266...
24  MULTIPOLYGON (((26514.58 28...
122 MULTIPOLYGON (((14483.48 31...
123 MULTIPOLYGON (((12861.38 32...
128 MULTIPOLYGON (((19680.06 31...
258 MULTIPOLYGON (((41343.11 40...
302 MULTIPOLYGON (((52567.43 46...
320 MULTIPOLYGON (((21693.06 48...

Notice that MPZ has 9 invalidity of sub zones here, so we have to make it valid through the function make valid. Once it’s valid we then check again

mpsz_sf <- st_make_valid(mpsz_sf)
mpsz_validity <- st_is_valid(mpsz_sf)
mpsz_invalid <- which(!mpsz_validity)
if (length(mpsz_invalid) > 0) {
  print("MPZ Invalid!")
  print(mpsz_sf[mpsz_invalid, ])
} else {
  print("it's valid!")
}
[1] "it's valid!"
sg_validity <- st_is_valid(sg_sf)
sg_invalid <- which(!sg_validity)
if (length(sg_invalid) > 0) {
  print("SG Invalid!")
  print(mpsz_sf[mpsz_invalid, ])
} else {
  print("it's valid!")
}
[1] "SG Invalid!"
Simple feature collection with 0 features and 15 fields
Bounding box:  xmin: NA ymin: NA xmax: NA ymax: NA
Projected CRS: SVY21 / Singapore TM
 [1] OBJECTID   SUBZONE_NO SUBZONE_N  SUBZONE_C  CA_IND     PLN_AREA_N
 [7] PLN_AREA_C REGION_N   REGION_C   INC_CRC    FMEL_UPD_D X_ADDR    
[13] Y_ADDR     SHAPE_Leng SHAPE_Area geometry  
<0 rows> (or 0-length row.names)

In SG_SF there’s one invalid as well, so we apply the fix.

sg_sf <- st_make_valid(sg_sf)
sg_validity <- st_is_valid(sg_sf)
sg_invalid <- which(!sg_validity)
if (length(sg_invalid) > 0) {
  print("SG Invalid!")
  print(mpsz_sf[sg_invalid, ])
} else {
  print("it's valid!")
}
[1] "it's valid!"

Notice that childcare is a geojson data and it houses it’s data in the description column, we need to break this up to get more meaningful data.

We can do a simple extraction from the Description attribute and map the data better. Assuming that each Table Row (TR) contains a Table Head (TH) and a Table Data (TD), we can map the data accordingly.

childcare_validity <- st_is_valid(childcare_sf)
childcare_invalid <- which(!childcare_validity)
if (length(childcare_invalid) > 0) {
  print("ChildCare Invalid!")
  print(childcare_sf[childcare_invalid, ])
} else {
  print("it's valid!")
}
[1] "it's valid!"
# Ensure the geometry column is preserved
geometry_column <- st_geometry(childcare_sf)
parse_description <- function(html_string) {
  html <- read_html(html_string)
  html <- html %>% html_nodes("tr") %>% .[!grepl("Attributes", .)]
  headers <- html %>% html_nodes("th") %>% html_text(trim = TRUE)
  values <- html %>% html_nodes("td") %>% html_text(trim = TRUE)
  
  # Handle cases where the number of headers and values don't match
  if (length(headers) != length(values)) {
    max_length <- max(length(headers), length(values))
    headers <- c(headers, rep("ExtraHeader", max_length - length(headers)))
    values <- c(values, rep("NULL", max_length - length(values)))
  }
  
  setNames(values, headers)
}

# Apply parsing function, unnest the description fields, and remove the original 'Description' column
childcare_sf <- childcare_sf %>% 
  mutate(Description_parsed = map(Description, parse_description)) %>%
  unnest_wider(Description_parsed) %>%
  select(-Description)  # Remove the original 'Description' column

# Overwrite the 'Name' column with the 'LANDYADDRESSPOINT' column values
childcare_sf <- childcare_sf %>%
  mutate(Name = NAME)  # Overwrite 'Name' with 'LANDYADDRESSPOINT'

# Replace empty strings or NA across all columns with "NULL"
childcare_sf <- childcare_sf %>%
  mutate(across(!geometry, ~ ifelse(is.na(.) | . == "", "NULL", .)))

# Reassign the geometry to the dataframe
st_geometry(childcare_sf) <- geometry_column
# Ensure it's still an sf object
class(childcare_sf)
[1] "sf"         "tbl_df"     "tbl"        "data.frame"

4.2 Mapping the geospatial datasets.

Using the mapping methods you learned in Hands-on Exercise 3, prepare a static map

# Suppress the tmap mode message
suppressMessages({
  tmap_mode("plot")  # Use "view" for an interactive map or "plot" for a static map
})

# Create the map
tm <- tm_shape(mpsz_sf) + 
  tm_polygons(col = "grey", border.col = "black", alpha = 0.5) +  # Base map with subzones
  tm_shape(childcare_sf) + 
  tm_dots(col = "black", size = 0.05) +  # Plot childcare locations as dots
   tm_layout(
    main.title = "Childcare Locations on Singapore Map",
    main.title.position = c("center"),  # Center the title at the top
    outer.margins = c(0.1, 0, 0, 0),  # Adjust outer margins to make space for the title
    legend.outside = TRUE,  # Keep legend outside the map area
    legend.outside.position = "bottom"  # Position the legend at the bottom
  )
tm

we can also prepare a pin map by using the code chunk below.

suppressMessages({
  tmap_mode("view")  # Use "view" for an interactive map or "plot" for a static map
})

tm <- tm_shape(mpsz_sf) + 
  tm_polygons(col = "grey", border.col = "black", alpha = 0.5) +  # Base map with subzones
  tm_shape(childcare_sf) + 
  tm_dots(col = "black", size = 0.05) +  # Plot childcare locations as dots
   tm_layout(
    title = "Childcare Locations on Singapore Map",
    title.position = c("center"),  # Center the title at the top
    outer.margins = c(0.1, 0, 0, 0),  # Adjust outer margins to make space for the title
    legend.outside = TRUE,  # Keep legend outside the map area
    legend.outside.position = "bottom"  # Position the legend at the bottom
  )

tm

5.0 Spatial Class Mapping

5.1 Data frame to Spatial Class

Use as_Spatial() to convert the data from dataframe to spatial class, we can check so using the class function or simply display it.

childcare <- as_Spatial(childcare_sf)
childcare
class       : SpatialPointsDataFrame 
features    : 1545 
extent      : 11203.01, 45404.24, 25667.6, 49300.88  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 16
names       :                    Name, ADDRESSBLOCKHOUSENUMBER, ADDRESSBUILDINGNAME, ADDRESSPOSTALCODE,                                                                       ADDRESSSTREETNAME, ADDRESSTYPE,         DESCRIPTION, HYPERLINK, LANDXADDRESSPOINT, LANDYADDRESSPOINT,                    NAME, PHOTOURL, ADDRESSFLOORNUMBER,          INC_CRC,     FMEL_UPD_D, ... 
min values  :    3-IN-1 FAMILY CENTRE,                    NULL,                NULL,            018989,                                                  1 & 3, Stratton Road, SINGAPORE 806787,        NULL, Child Care Services,      NULL,                 0,                 0,    3-IN-1 FAMILY CENTRE,     NULL,               NULL, 00A958622500BF89, 20200812221033, ... 
max values  : ZEE SCHOOLHOUSE PTE LTD,                    NULL,                NULL,            829646, UPPER BASEMENT LEVEL, WEST WING, TERMINAL 1, SINGAPORE CHANGI AIRPORT, SINGAPORE 819642,        NULL,                NULL,      NULL,                 0,                 0, ZEE SCHOOLHOUSE PTE LTD,     NULL,               NULL, FFCFA88A8CE5665A, 20200826094036, ... 
class(childcare)
[1] "SpatialPointsDataFrame"
attr(,"package")
[1] "sp"
mpsz <- as_Spatial(mpsz_sf)
mpsz
class       : SpatialPolygonsDataFrame 
features    : 323 
extent      : 2667.538, 56396.44, 15748.72, 50256.33  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 15
names       : OBJECTID, SUBZONE_NO, SUBZONE_N, SUBZONE_C, CA_IND, PLN_AREA_N, PLN_AREA_C,       REGION_N, REGION_C,          INC_CRC, FMEL_UPD_D,     X_ADDR,     Y_ADDR,    SHAPE_Leng,    SHAPE_Area 
min values  :        1,          1, ADMIRALTY,    AMSZ01,      N, ANG MO KIO,         AM, CENTRAL REGION,       CR, 00F5E30B5C9B7AD8,      16409,  5092.8949,  19579.069, 871.554887798, 39437.9352703 
max values  :      323,         17,    YUNNAN,    YSSZ09,      Y,     YISHUN,         YS,    WEST REGION,       WR, FFCCF172717C2EAF,      16409, 50424.7923, 49552.7904, 68083.9364708,  69748298.792 
class(mpsz)
[1] "SpatialPolygonsDataFrame"
attr(,"package")
[1] "sp"
sg <- as_Spatial(sg_sf)
sg
class       : SpatialPolygonsDataFrame 
features    : 60 
extent      : 2663.926, 56047.79, 16357.98, 50244.03  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 4
names       : GDO_GID, MSLINK, MAPID,              COSTAL_NAM 
min values  :       1,      1,     0,             ISLAND LINK 
max values  :      60,     67,     0, SINGAPORE - MAIN ISLAND 
class(sg)
[1] "SpatialPolygonsDataFrame"
attr(,"package")
[1] "sp"

5.2 Converting Spatial Class into generic PPP Format

As Spatstat requires analytical data in ppp object form. We have to map the data to a PPP object. The following steps breakdown the method to convert a SF to PPP object.

  • Extract Coordinates (st_coordinates(childcare_sf)):
    This step extracts the x (longitude) and y (latitude) coordinates from the sf object. The result is a matrix with two columns—one for each coordinate. These coordinates represent the location of each point in our spatial dataset.
  • Get Bounding Box (st_bbox(childcare_sf)):
    This function retrieves the bounding box of the sf object, which is the smallest rectangle that can enclose all the points in the dataset. The bounding box provides the minimum and maximum x and y values (xmin, xmax, ymin, ymax)

  • Create Observation Window (owin()):
    Using the bounding box values, you create an observation window. This window defines the spatial limits (study area) for the point pattern analysis. It ensures that all points lie within these specified boundaries.

  • Create ppp Object (ppp()):
    Finally, we combine the extracted coordinates and the defined observation window into a ppp object using the ppp() function. The ppp object is the required format for analyzing point patterns in the spatstat package, enabling us to conduct spatial analyses on our data.

# Extract coordinates
childcare_coords <- st_coordinates(childcare_sf)

# Define the window using the bounding box
childcare_bbox <- st_bbox(childcare_sf)
childcare_window <- owin(xrange = childcare_bbox[c("xmin", "xmax")], yrange = childcare_bbox[c("ymin", "ymax")])

# Create the ppp object
childcare_ppp <- ppp(x = childcare_coords[, 1], y = childcare_coords[, 2], window = childcare_window)

# Check the ppp object
summary(childcare_ppp)
Planar point pattern:  1545 points
Average intensity 1.91145e-06 points per square unit

*Pattern contains duplicated points*

Coordinates are given to 11 decimal places

Window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
                    (34200 x 23630 units)
Window area = 808287000 square units
plot(childcare_ppp)

5.3 Checking for duplicate data.

5.3.1 Code to analyse for duplicate data.

any(duplicated(childcare_ppp))
[1] TRUE
multiplicity(childcare_ppp)
   1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16 
   1    1    1    3    1    1    1    1    2    1    1    1    1    1    1    1 
  17   18   19   20   21   22   23   24   25   26   27   28   29   30   31   32 
   1    1    1    1    1    1    1    1    1    1    9    1    1    1    1    1 
  33   34   35   36   37   38   39   40   41   42   43   44   45   46   47   48 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
  49   50   51   52   53   54   55   56   57   58   59   60   61   62   63   64 
   1    1    1    1    1    1    2    1    1    3    1    1    1    1    1    1 
  65   66   67   68   69   70   71   72   73   74   75   76   77   78   79   80 
   1    1    1    1    1    2    1    1    1    1    1    2    1    1    1    1 
  81   82   83   84   85   86   87   88   89   90   91   92   93   94   95   96 
   1    1    1    3    1    1    1    1    1    1    1    1    1    1    1    1 
  97   98   99  100  101  102  103  104  105  106  107  108  109  110  111  112 
   1    1    1    1    1    1    1    1    2    1    1    1    1    1    1    1 
 113  114  115  116  117  118  119  120  121  122  123  124  125  126  127  128 
   1    1    1    1    1    1    2    1    1    1    3    1    1    1    2    1 
 129  130  131  132  133  134  135  136  137  138  139  140  141  142  143  144 
   1    1    1    1    1    2    1    1    1    1    1    1    1    1    3    2 
 145  146  147  148  149  150  151  152  153  154  155  156  157  158  159  160 
   1    2    1    1    1    2    2    3    1    5    1    5    1    1    1    2 
 161  162  163  164  165  166  167  168  169  170  171  172  173  174  175  176 
   1    1    1    1    2    1    1    1    1    1    1    2    1    1    1    1 
 177  178  179  180  181  182  183  184  185  186  187  188  189  190  191  192 
   1    4    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 193  194  195  196  197  198  199  200  201  202  203  204  205  206  207  208 
   1    1    1    1    1    2    2    1    1    1    1    2    1    4    1    1 
 209  210  211  212  213  214  215  216  217  218  219  220  221  222  223  224 
   2    1    1    1    1    1    1    1    1    1    1    1    2    1    1    1 
 225  226  227  228  229  230  231  232  233  234  235  236  237  238  239  240 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 241  242  243  244  245  246  247  248  249  250  251  252  253  254  255  256 
   1    1    1    1    2    1    1    1    1    1    1    1    1    1    1    1 
 257  258  259  260  261  262  263  264  265  266  267  268  269  270  271  272 
   1    1    1    1    1    1    1    1    1    1    2    1    1    1    1    3 
 273  274  275  276  277  278  279  280  281  282  283  284  285  286  287  288 
   1    1    1    1    1    1    3    1    1    1    1    1    1    1    1    1 
 289  290  291  292  293  294  295  296  297  298  299  300  301  302  303  304 
   1    1    1    1    1    1    1    9    1    1    2    1    1    1    1    1 
 305  306  307  308  309  310  311  312  313  314  315  316  317  318  319  320 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 321  322  323  324  325  326  327  328  329  330  331  332  333  334  335  336 
   1    1    1    5    1    1    1    1    1    2    1    1    2    2    1    1 
 337  338  339  340  341  342  343  344  345  346  347  348  349  350  351  352 
   1    1    1    1    1    1    1    1    1    1    1    1    1    2    2    1 
 353  354  355  356  357  358  359  360  361  362  363  364  365  366  367  368 
   1    1    1    1    9    1    1    1    1    1    1    1    1    1    1    1 
 369  370  371  372  373  374  375  376  377  378  379  380  381  382  383  384 
   1    3    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 385  386  387  388  389  390  391  392  393  394  395  396  397  398  399  400 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 401  402  403  404  405  406  407  408  409  410  411  412  413  414  415  416 
   1    1    2    1    1    1    1    1    1    1    2    1    1    1    1    1 
 417  418  419  420  421  422  423  424  425  426  427  428  429  430  431  432 
   1    1    1    1    1    1    1    2    1    1    2    1    1    1    1    1 
 433  434  435  436  437  438  439  440  441  442  443  444  445  446  447  448 
   1    1    1    1    2    1    1    1    1    1    1    1    1    1    1    1 
 449  450  451  452  453  454  455  456  457  458  459  460  461  462  463  464 
   1    1    9    9    1    1    1    1    1    1    1    1    1    1    2    1 
 465  466  467  468  469  470  471  472  473  474  475  476  477  478  479  480 
   2    1    1    1    1    1    1    1    1    1    1    1    2    2    1    1 
 481  482  483  484  485  486  487  488  489  490  491  492  493  494  495  496 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 497  498  499  500  501  502  503  504  505  506  507  508  509  510  511  512 
   1    1    1    1    1    1    2    1    1    1    1    1    1    1    1    2 
 513  514  515  516  517  518  519  520  521  522  523  524  525  526  527  528 
   1    1    1    1    1    1    1    1    1    1    1    2    1    1    3    1 
 529  530  531  532  533  534  535  536  537  538  539  540  541  542  543  544 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 545  546  547  548  549  550  551  552  553  554  555  556  557  558  559  560 
   1    1    1    1    1    1    1    1    1    3    1    1    1    1    1    1 
 561  562  563  564  565  566  567  568  569  570  571  572  573  574  575  576 
   2    2    2    1    1    1    1    2    1    1    2    1    1    1    2    1 
 577  578  579  580  581  582  583  584  585  586  587  588  589  590  591  592 
   1    2    1    1    1    1    1    9    1    4    1    2    1    1    1    1 
 593  594  595  596  597  598  599  600  601  602  603  604  605  606  607  608 
   2    1    1    1    1    1    1    1    2    1    2    1    1    1    1    1 
 609  610  611  612  613  614  615  616  617  618  619  620  621  622  623  624 
   1    1    1    1    1    1    1    1    1    2    1    2    1    1    1    1 
 625  626  627  628  629  630  631  632  633  634  635  636  637  638  639  640 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 641  642  643  644  645  646  647  648  649  650  651  652  653  654  655  656 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    4 
 657  658  659  660  661  662  663  664  665  666  667  668  669  670  671  672 
   1    1    1    1    1    1    1    3    1    1    1    1    1    1    1    1 
 673  674  675  676  677  678  679  680  681  682  683  684  685  686  687  688 
   1    1    1    1    1    4    1    1    1    1    1    4    1    1    1    1 
 689  690  691  692  693  694  695  696  697  698  699  700  701  702  703  704 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 705  706  707  708  709  710  711  712  713  714  715  716  717  718  719  720 
   1    1    2    1    1    1    1    1    1    1    1    1    1    1    1    1 
 721  722  723  724  725  726  727  728  729  730  731  732  733  734  735  736 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 737  738  739  740  741  742  743  744  745  746  747  748  749  750  751  752 
   1    2    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 753  754  755  756  757  758  759  760  761  762  763  764  765  766  767  768 
   1    1    1    1    1    2    1    1    1    1    1    1    1    1    1    1 
 769  770  771  772  773  774  775  776  777  778  779  780  781  782  783  784 
   1    1    1    1    1    1    1    1    1    4    1    1    1    1    1    1 
 785  786  787  788  789  790  791  792  793  794  795  796  797  798  799  800 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 801  802  803  804  805  806  807  808  809  810  811  812  813  814  815  816 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 817  818  819  820  821  822  823  824  825  826  827  828  829  830  831  832 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 833  834  835  836  837  838  839  840  841  842  843  844  845  846  847  848 
   1    1    1    1    1    1    1    2    1    1    1    1    1    1    1    1 
 849  850  851  852  853  854  855  856  857  858  859  860  861  862  863  864 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 865  866  867  868  869  870  871  872  873  874  875  876  877  878  879  880 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    2 
 881  882  883  884  885  886  887  888  889  890  891  892  893  894  895  896 
   3    1    1    1    2    1    1    1    3    1    1    3    1    1    1    1 
 897  898  899  900  901  902  903  904  905  906  907  908  909  910  911  912 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 913  914  915  916  917  918  919  920  921  922  923  924  925  926  927  928 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 929  930  931  932  933  934  935  936  937  938  939  940  941  942  943  944 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 945  946  947  948  949  950  951  952  953  954  955  956  957  958  959  960 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    2 
 961  962  963  964  965  966  967  968  969  970  971  972  973  974  975  976 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 977  978  979  980  981  982  983  984  985  986  987  988  989  990  991  992 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 993  994  995  996  997  998  999 1000 1001 1002 1003 1004 1005 1006 1007 1008 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 
   1    1    1    1    1    1    1    1    1    2    2    1    1    1    1    1 
1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 
   1    1    1    1    1    2    1    1    1    1    1    1    1    1    1    1 
1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 
   1    1    1    1    1    1    1    1    2    2    1    1    1    5    1    1 
1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 
   1    1    1    1    1    1    1    1    1    2    1    1    1    1    1    1 
1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 
   1    1    1    1    1    1    1    1    1    1    2    1    1    1    1    1 
1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 
   1    9    1    2    2    1    1    1    2    1    1    1    1    1    1    1 
1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 
   1    1    1    1    2    1    1    1    3    1    1    1    1    1    1    1 
1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 
   9    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 
   1    1    1    2    1    1    1    1    1    1    1    1    1    1    1    1 
1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    2 
1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 
   1    1    1    2    1    2    1    1    1    2    2    2    1    1    1    1 
1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 
   1    1    2    1    1    1    1    1    1    1    1    1    2    1    1    1 
1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 
   1    1    1    1    3    1    1    1    1    1    1    1    1    1    1    1 
1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 
   1    1    1    1    1    1    1    1    4    1    1    1    1    1    2    1 
1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 
   1    1    1    1    1    1    1    1    1    9    1    1    1    1    1    1 
1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    2    1 
1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 
   1    2    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 
   1    1    1    1    1    1    1    1    1    1    2    1    1    1    1    1 
1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 
   1    1    1    1    1    1    2    1    1    1    1    1    1    1    1    1 
1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 
   1    1    1    1    1    1    1    1    1    1    5    1    1    1    1    1 
1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 
   1    1    1    1    1    2    1    1    1    1    2    1    1    1    1    3 
1537 1538 1539 1540 1541 1542 1543 1544 1545 
   1    1    1    1    1    1    2    1    1 
sum(multiplicity(childcare_ppp) > 1)
[1] 128

5.3.2 Spot duplicate points from the map

# Identify duplicates in the ppp object
childcare_duplicate_indices <- duplicated(childcare_ppp)
# Extract the coordinates of duplicate points
childcare_duplicate_coords <- childcare_ppp[childcare_duplicate_indices]
# Plot the original points
plot(childcare_ppp, main = "Childcare Locations with Duplicate Points Highlighted")
# Overlay duplicate points in a different color
points(childcare_duplicate_coords$x, childcare_duplicate_coords$y, col = "red", pch = 19, cex = 0.7)

5.4 Handling Duplicates Events.

5.4.1 Methods to Handle Duplicates

Three Methods

  • Deleting Duplicates (unique_childcare_ppp): Removes duplicate points, resulting in a dataset with only unique events.
  • Jittering Duplicates (jittered_childcare_ppp): Slightly perturbs duplicate points to distinguish them spatially, preventing them from overlapping completely.

  • Unique Marks (marked_childcare_ppp): Attaches a “mark” to each point, especially duplicates, which can be used later in the analysis to account for the fact that these points were originally duplicates.

unique_childcare_ppp <- childcare_ppp[!duplicated(childcare_ppp)]

# Check the ppp object after removing duplicates
summary(unique_childcare_ppp)
Planar point pattern:  1471 points
Average intensity 1.819898e-06 points per square unit

Coordinates are given to 11 decimal places

Window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
                    (34200 x 23630 units)
Window area = 808287000 square units
plot(unique_childcare_ppp, main = "Childcare Locations Without Duplicates")

# Jitter the coordinates to handle duplicates
jittered_coords <- childcare_coords
jittered_coords[duplicated(childcare_ppp), ] <- jitter(jittered_coords[duplicated(childcare_ppp), ], amount = 0.01)

# Create a new ppp object with jittered points
jittered_childcare_ppp <- ppp(x = jittered_coords[, 1], y = jittered_coords[, 2], window = childcare_window)

# Check the ppp object after jittering
summary(jittered_childcare_ppp)
Planar point pattern:  1545 points
Average intensity 1.91145e-06 points per square unit

Coordinates are given to 11 decimal places

Window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
                    (34200 x 23630 units)
Window area = 808287000 square units
plot(jittered_childcare_ppp, main = "Childcare Locations with Jittered Duplicates")

# Create marks for duplicates
marks <- rep(1, npoints(childcare_ppp))
marks[duplicated(childcare_ppp)] <- 2

# Create a new ppp object with marks attached to each point
marked_childcare_ppp <- ppp(x = childcare_coords[, 1], y = childcare_coords[, 2], window = childcare_window, marks = marks)

# Check the ppp object with unique marks
summary(marked_childcare_ppp)
Marked planar point pattern:  1545 points
Average intensity 1.91145e-06 points per square unit

*Pattern contains duplicated points*

Coordinates are given to 11 decimal places

marks are numeric, of type 'double'
Summary:
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  1.000   1.000   1.000   1.048   1.000   2.000 

Window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
                    (34200 x 23630 units)
Window area = 808287000 square units
plot(marked_childcare_ppp, main = "Childcare Locations with Unique Marks", cols = c("black", "red"))

5.4.2 Last check for duplicates

any(duplicated(unique_childcare_ppp))
[1] FALSE
any(duplicated(jittered_childcare_ppp))
[1] FALSE
any(duplicated(marked_childcare_ppp))
[1] TRUE

Notice that mark will still make it true, because there are still duplicates but marked differently.

I would use jittered from here forward

5.5 Creating OWIN Object

OWIN is used to represent the polygonal region, and we us the SG_SF to plot the map.

sg_owin <- as.owin(sg_sf)
plot(sg_owin)

summary(sg_owin)
Window: polygonal boundary
51 separate polygons (2 holes)
                  vertices         area relative.area
polygon 1 (hole)        30 -7.08118e+03     -9.76e-06
polygon 2               55  8.25379e+04      1.14e-04
polygon 3               90  4.15092e+05      5.72e-04
polygon 4               49  1.66986e+04      2.30e-05
polygon 5               38  2.42492e+04      3.34e-05
polygon 6              976  2.33447e+07      3.22e-02
polygon 7              721  1.92795e+06      2.66e-03
polygon 8             1989  9.99217e+06      1.38e-02
polygon 9              330  1.11896e+06      1.54e-03
polygon 10             175  9.25904e+05      1.28e-03
polygon 11             115  9.28394e+05      1.28e-03
polygon 12              24  6.35239e+03      8.76e-06
polygon 13 (hole)        3 -1.06765e+00     -1.47e-09
polygon 14             190  2.02489e+05      2.79e-04
polygon 15              37  1.01705e+04      1.40e-05
polygon 16              25  1.66227e+04      2.29e-05
polygon 17              10  2.14507e+03      2.96e-06
polygon 18              66  1.61841e+04      2.23e-05
polygon 19            5195  6.36837e+08      8.78e-01
polygon 20              76  3.12332e+05      4.31e-04
polygon 21             627  3.18913e+07      4.40e-02
polygon 22              20  3.28420e+04      4.53e-05
polygon 23              42  5.58317e+04      7.70e-05
polygon 24              67  1.31354e+06      1.81e-03
polygon 25             734  4.69093e+06      6.47e-03
polygon 26              16  3.19460e+03      4.40e-06
polygon 27              15  4.87296e+03      6.72e-06
polygon 28              15  4.46420e+03      6.15e-06
polygon 29              14  5.46674e+03      7.54e-06
polygon 30              37  5.26194e+03      7.25e-06
polygon 31             111  6.62927e+05      9.14e-04
polygon 32              69  5.63134e+04      7.76e-05
polygon 33             143  1.45139e+05      2.00e-04
polygon 34             397  2.48821e+06      3.43e-03
polygon 35              90  1.15991e+05      1.60e-04
polygon 36              98  6.26829e+04      8.64e-05
polygon 37             165  3.38736e+05      4.67e-04
polygon 38             130  9.40465e+04      1.30e-04
polygon 39              93  4.30642e+05      5.94e-04
polygon 40              16  2.01046e+03      2.77e-06
polygon 41             415  3.25384e+06      4.49e-03
polygon 42              30  1.08382e+04      1.49e-05
polygon 43              53  3.44003e+04      4.74e-05
polygon 44              26  8.34758e+03      1.15e-05
polygon 45              74  5.82234e+04      8.03e-05
polygon 46             327  2.16921e+06      2.99e-03
polygon 47             177  4.67446e+05      6.44e-04
polygon 48              46  6.99702e+05      9.65e-04
polygon 49               6  1.68410e+04      2.32e-05
polygon 50              13  7.00873e+04      9.66e-05
polygon 51               4  9.45963e+03      1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401

5.6 Combining Point events object and owin object

Extract and combine the point and polygon feaature in one ppp object.

childcareSG_ppp = jittered_childcare_ppp[sg_owin]
summary(childcareSG_ppp)
Planar point pattern:  1545 points
Average intensity 2.129929e-06 points per square unit

Coordinates are given to 11 decimal places

Window: polygonal boundary
51 separate polygons (2 holes)
                  vertices         area relative.area
polygon 1 (hole)        30 -7.08118e+03     -9.76e-06
polygon 2               55  8.25379e+04      1.14e-04
polygon 3               90  4.15092e+05      5.72e-04
polygon 4               49  1.66986e+04      2.30e-05
polygon 5               38  2.42492e+04      3.34e-05
polygon 6              976  2.33447e+07      3.22e-02
polygon 7              721  1.92795e+06      2.66e-03
polygon 8             1989  9.99217e+06      1.38e-02
polygon 9              330  1.11896e+06      1.54e-03
polygon 10             175  9.25904e+05      1.28e-03
polygon 11             115  9.28394e+05      1.28e-03
polygon 12              24  6.35239e+03      8.76e-06
polygon 13 (hole)        3 -1.06765e+00     -1.47e-09
polygon 14             190  2.02489e+05      2.79e-04
polygon 15              37  1.01705e+04      1.40e-05
polygon 16              25  1.66227e+04      2.29e-05
polygon 17              10  2.14507e+03      2.96e-06
polygon 18              66  1.61841e+04      2.23e-05
polygon 19            5195  6.36837e+08      8.78e-01
polygon 20              76  3.12332e+05      4.31e-04
polygon 21             627  3.18913e+07      4.40e-02
polygon 22              20  3.28420e+04      4.53e-05
polygon 23              42  5.58317e+04      7.70e-05
polygon 24              67  1.31354e+06      1.81e-03
polygon 25             734  4.69093e+06      6.47e-03
polygon 26              16  3.19460e+03      4.40e-06
polygon 27              15  4.87296e+03      6.72e-06
polygon 28              15  4.46420e+03      6.15e-06
polygon 29              14  5.46674e+03      7.54e-06
polygon 30              37  5.26194e+03      7.25e-06
polygon 31             111  6.62927e+05      9.14e-04
polygon 32              69  5.63134e+04      7.76e-05
polygon 33             143  1.45139e+05      2.00e-04
polygon 34             397  2.48821e+06      3.43e-03
polygon 35              90  1.15991e+05      1.60e-04
polygon 36              98  6.26829e+04      8.64e-05
polygon 37             165  3.38736e+05      4.67e-04
polygon 38             130  9.40465e+04      1.30e-04
polygon 39              93  4.30642e+05      5.94e-04
polygon 40              16  2.01046e+03      2.77e-06
polygon 41             415  3.25384e+06      4.49e-03
polygon 42              30  1.08382e+04      1.49e-05
polygon 43              53  3.44003e+04      4.74e-05
polygon 44              26  8.34758e+03      1.15e-05
polygon 45              74  5.82234e+04      8.03e-05
polygon 46             327  2.16921e+06      2.99e-03
polygon 47             177  4.67446e+05      6.44e-04
polygon 48              46  6.99702e+05      9.65e-04
polygon 49               6  1.68410e+04      2.32e-05
polygon 50              13  7.00873e+04      9.66e-05
polygon 51               4  9.45963e+03      1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401

Plot the map as shown below here by:

plot(childcareSG_ppp)

6.0 First-Order Spatial Point Pattern Analysis.

In this section, wewill learn how to perform first-order SPPA by using spatstat package. The hands-on exercise will focus on:

  • deriving kernel density estimation (KDE) layer for visualising and exploring the intensity of point processes,

  • performing Confirmatory Spatial Point Patterns Analysis by using Nearest Neighbour statistics.

6.1 Kernel Density Estimation

6.1.1 Understanding KDE

Kernel Density Estimation (KDE) is a non-parametric way to estimate the probability density function of a random variable. In spatial analysis, KDE is used to estimate the intensity of point patterns across a study area, which helps to identify hotspots or areas with high concentrations of events (e.g., childcare locations).

Steps in KDE:

  1. Kernel Function:

    • The kernel function is a smooth, symmetric function (often Gaussian) that is used to estimate the density at each point. It determines how much influence each point has on the surrounding area.

    • In the spatial context, each point in your dataset contributes to the density estimate, with its influence decreasing with distance according to the kernel function.

  2. Bandwidth (sigma):

    • The bandwidth parameter (sigma) controls the width of the kernel function. It determines the scale of smoothing:

      • Small bandwidth: Results in a more detailed map with sharper peaks but may be too sensitive to noise.

      • Large bandwidth: Produces a smoother map but may oversmooth the data, losing important details.

    • Bandwidth selection is crucial for accurate density estimation. One common method for selecting bandwidth is Diggle’s bandwidth (bw.diggle), which is specifically designed for spatial point patterns.

  3. Edge Correction:

    • When performing KDE on finite study areas, edge effects can bias the density estimates near the boundaries.

    • Edge correction (edge=TRUE) adjusts for this by accounting for the missing density outside the boundaries, leading to more accurate results near the edges.

  4. Density Calculation:

    • The KDE produces a continuous surface (usually a raster or grid) where each cell represents the estimated density of points. Higher values indicate areas with a higher concentration of points.

6.1.2 As Seen in Code

Breaking Down the Code:

  1. density(childcareSG_ppp, ...):

    • This function from the spatstat package computes the Kernel Density Estimation (KDE) for the ppp object childcareSG_ppp.
  2. sigma=bw.diggle:

    • sigma specifies the bandwidth (smoothing parameter). Here, bw.diggle is used to automatically calculate the optimal bandwidth based on Diggle’s method, which balances the trade-off between detail and smoothness.
  3. edge=TRUE:

    • This argument enables edge correction, adjusting the density estimate near the boundaries of the study area to avoid underestimation due to the edge effect.
  4. kernel="gaussian":

    • Specifies the type of kernel function to use. The Gaussian kernel is the most commonly used, providing a smooth, bell-shaped curve that smoothly decreases in influence as distance from the point increases.
kde_childcareSG_bw <- density(childcareSG_ppp,
                              sigma=bw.diggle,
                              edge=TRUE,
                            kernel="gaussian") 
plot(kde_childcareSG_bw)

Retrieving the bandwidth to compute the kde layer

bw <- bw.diggle(childcareSG_ppp)
bw
   sigma 
298.4095 

6.1.3 Rescaling KDE Values

summary(childcareSG_ppp)
Planar point pattern:  1545 points
Average intensity 2.129929e-06 points per square unit

Coordinates are given to 11 decimal places

Window: polygonal boundary
51 separate polygons (2 holes)
                  vertices         area relative.area
polygon 1 (hole)        30 -7.08118e+03     -9.76e-06
polygon 2               55  8.25379e+04      1.14e-04
polygon 3               90  4.15092e+05      5.72e-04
polygon 4               49  1.66986e+04      2.30e-05
polygon 5               38  2.42492e+04      3.34e-05
polygon 6              976  2.33447e+07      3.22e-02
polygon 7              721  1.92795e+06      2.66e-03
polygon 8             1989  9.99217e+06      1.38e-02
polygon 9              330  1.11896e+06      1.54e-03
polygon 10             175  9.25904e+05      1.28e-03
polygon 11             115  9.28394e+05      1.28e-03
polygon 12              24  6.35239e+03      8.76e-06
polygon 13 (hole)        3 -1.06765e+00     -1.47e-09
polygon 14             190  2.02489e+05      2.79e-04
polygon 15              37  1.01705e+04      1.40e-05
polygon 16              25  1.66227e+04      2.29e-05
polygon 17              10  2.14507e+03      2.96e-06
polygon 18              66  1.61841e+04      2.23e-05
polygon 19            5195  6.36837e+08      8.78e-01
polygon 20              76  3.12332e+05      4.31e-04
polygon 21             627  3.18913e+07      4.40e-02
polygon 22              20  3.28420e+04      4.53e-05
polygon 23              42  5.58317e+04      7.70e-05
polygon 24              67  1.31354e+06      1.81e-03
polygon 25             734  4.69093e+06      6.47e-03
polygon 26              16  3.19460e+03      4.40e-06
polygon 27              15  4.87296e+03      6.72e-06
polygon 28              15  4.46420e+03      6.15e-06
polygon 29              14  5.46674e+03      7.54e-06
polygon 30              37  5.26194e+03      7.25e-06
polygon 31             111  6.62927e+05      9.14e-04
polygon 32              69  5.63134e+04      7.76e-05
polygon 33             143  1.45139e+05      2.00e-04
polygon 34             397  2.48821e+06      3.43e-03
polygon 35              90  1.15991e+05      1.60e-04
polygon 36              98  6.26829e+04      8.64e-05
polygon 37             165  3.38736e+05      4.67e-04
polygon 38             130  9.40465e+04      1.30e-04
polygon 39              93  4.30642e+05      5.94e-04
polygon 40              16  2.01046e+03      2.77e-06
polygon 41             415  3.25384e+06      4.49e-03
polygon 42              30  1.08382e+04      1.49e-05
polygon 43              53  3.44003e+04      4.74e-05
polygon 44              26  8.34758e+03      1.15e-05
polygon 45              74  5.82234e+04      8.03e-05
polygon 46             327  2.16921e+06      2.99e-03
polygon 47             177  4.67446e+05      6.44e-04
polygon 48              46  6.99702e+05      9.65e-04
polygon 49               6  1.68410e+04      2.32e-05
polygon 50              13  7.00873e+04      9.66e-05
polygon 51               4  9.45963e+03      1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401
childcareSG_ppp.km <- rescale.ppp(childcareSG_ppp, 1000, "km")
summary(childcareSG_ppp.km)
Planar point pattern:  1545 points
Average intensity 2.129929 points per square km

Coordinates are given to 14 decimal places

Window: polygonal boundary
51 separate polygons (2 holes)
                  vertices         area relative.area
polygon 1 (hole)        30 -7.08118e-03     -9.76e-06
polygon 2               55  8.25379e-02      1.14e-04
polygon 3               90  4.15092e-01      5.72e-04
polygon 4               49  1.66986e-02      2.30e-05
polygon 5               38  2.42492e-02      3.34e-05
polygon 6              976  2.33447e+01      3.22e-02
polygon 7              721  1.92795e+00      2.66e-03
polygon 8             1989  9.99217e+00      1.38e-02
polygon 9              330  1.11896e+00      1.54e-03
polygon 10             175  9.25904e-01      1.28e-03
polygon 11             115  9.28394e-01      1.28e-03
polygon 12              24  6.35239e-03      8.76e-06
polygon 13 (hole)        3 -1.06765e-06     -1.47e-09
polygon 14             190  2.02489e-01      2.79e-04
polygon 15              37  1.01705e-02      1.40e-05
polygon 16              25  1.66227e-02      2.29e-05
polygon 17              10  2.14507e-03      2.96e-06
polygon 18              66  1.61841e-02      2.23e-05
polygon 19            5195  6.36837e+02      8.78e-01
polygon 20              76  3.12332e-01      4.31e-04
polygon 21             627  3.18913e+01      4.40e-02
polygon 22              20  3.28420e-02      4.53e-05
polygon 23              42  5.58317e-02      7.70e-05
polygon 24              67  1.31354e+00      1.81e-03
polygon 25             734  4.69093e+00      6.47e-03
polygon 26              16  3.19460e-03      4.40e-06
polygon 27              15  4.87296e-03      6.72e-06
polygon 28              15  4.46420e-03      6.15e-06
polygon 29              14  5.46674e-03      7.54e-06
polygon 30              37  5.26194e-03      7.25e-06
polygon 31             111  6.62927e-01      9.14e-04
polygon 32              69  5.63134e-02      7.76e-05
polygon 33             143  1.45139e-01      2.00e-04
polygon 34             397  2.48821e+00      3.43e-03
polygon 35              90  1.15991e-01      1.60e-04
polygon 36              98  6.26829e-02      8.64e-05
polygon 37             165  3.38736e-01      4.67e-04
polygon 38             130  9.40465e-02      1.30e-04
polygon 39              93  4.30642e-01      5.94e-04
polygon 40              16  2.01046e-03      2.77e-06
polygon 41             415  3.25384e+00      4.49e-03
polygon 42              30  1.08382e-02      1.49e-05
polygon 43              53  3.44003e-02      4.74e-05
polygon 44              26  8.34758e-03      1.15e-05
polygon 45              74  5.82234e-02      8.03e-05
polygon 46             327  2.16921e+00      2.99e-03
polygon 47             177  4.67446e-01      6.44e-04
polygon 48              46  6.99702e-01      9.65e-04
polygon 49               6  1.68410e-02      2.32e-05
polygon 50              13  7.00873e-02      9.66e-05
polygon 51               4  9.45963e-03      1.30e-05
enclosing rectangle: [2.66393, 56.04779] x [16.35798, 50.24403] km
                     (53.38 x 33.89 km)
Window area = 725.376 square km
Unit of length: 1 km
Fraction of frame area: 0.401
kde_childcareSG.bw <- density(childcareSG_ppp.km, sigma=bw.diggle, edge=TRUE, kernel="gaussian")
plot(kde_childcareSG.bw,main = "Kernel Density Estimation of Childcare Locations (Rescaled to KM)")

6.2 working with different automatic bandwidth methods

Bandwidth is a crucial parameter in Kernel Density Estimation (KDE). It controls the degree of smoothing applied to the data. Different methods for selecting the bandwidth lead to different levels of smoothing, which can impact the interpretation of the density estimate.

In the spatstat package, several functions are available to determine the optimal bandwidth for KDE:

  1. bw.diggle():

    • Purpose: Designed for spatial point patterns, it aims to balance the trade-off between bias and variance in the density estimate.

    • Characteristics: Often produces a good balance between under- and over-smoothing, making it suitable for general spatial analysis.

  2. bw.CvL() (Cronie and Van Lieshout):

    • Purpose: This method minimizes the integrated squared error between the true intensity function and the estimated intensity function.

    • Characteristics: It’s particularly good for minimizing error over the entire study area, but it can be sensitive to the overall distribution of points.

  3. bw.scott() (Scott’s Rule):

    • Purpose: Based on Scott’s rule of thumb, this method provides a bandwidth that scales with the number of points and the dimension of the data.

    • Characteristics: Often results in a conservative (wider) bandwidth, leading to smoother density estimates that may miss finer details.

  4. bw.ppl() (Likelihood Cross-Validation):

    • Purpose: This method uses cross-validation to select a bandwidth that maximizes the likelihood of the observed data under the KDE model.

    • Characteristics: It tends to focus on how well the bandwidth explains the data, often leading to a finer, more detailed density estimate.

bw.CvL(childcareSG_ppp.km)
   sigma 
2.562099 
bw.scott(childcareSG_ppp.km)
 sigma.x  sigma.y 
2.224898 1.450966 
bw.ppl(childcareSG_ppp.km)
    sigma 
0.2290091 
bw.diggle(childcareSG_ppp.km)
    sigma 
0.2984095 
# Perform Kernel Density Estimation with different bandwidth selection methods
kde_childcareSG.ppl <- density(childcareSG_ppp.km, sigma=bw.ppl, edge=TRUE, kernel="gaussian")
kde_childcareSG.scott <- density(childcareSG_ppp.km, sigma=bw.scott, edge=TRUE, kernel="gaussian")
kde_childcareSG.CvL <- density(childcareSG_ppp.km, sigma=bw.CvL, edge=TRUE, kernel="gaussian")
kde_childcareSG.diggle <- density(childcareSG_ppp.km, sigma=bw.diggle, edge=TRUE, kernel="gaussian")


layout(matrix(c(1, 2, 3, 4), 2, 2, byrow = TRUE), widths = c(1, 1), heights = c(1, 1))
# Set margins to the minimum (bottom, left, top, right)
par(mar = c(2, 2, 2, 2), oma = c(0, 0, 0, 0))
# Plot the results for comparison
par(mfrow=c(2,2))  # Arrange plots in a 2x2 grid
plot(kde_childcareSG.diggle, main = "bw.diggle")
plot(kde_childcareSG.ppl, main = "bw.ppl")
plot(kde_childcareSG.CvL, main = "bw.CvL")
plot(kde_childcareSG.scott, main = "bw.scott")

6.3 working with different kernel methods.

Kernel Method Shape Characterisitics
Gaussian Bell shaped- normal Very smooth, widely used, good for most applications, but might oversmooth and miss finer details.
Epanechnikov Parabolic Efficient, minimizes estimation error, compact support (affects nearby points), less smooth than Gaussian.
Quartic Bell Shape with flat top Balanced smoothness and efficiency, compact support, focuses on nearby points, similar to Epanechnikov.
Uniform Rectangle Simple and fast, gives equal weight within a certain distance, but produces rougher estimates.
layout(matrix(c(1, 2, 3, 4), 2, 2, byrow = TRUE), widths = c(1, 1), heights = c(1, 1))

# Set margins to the minimum (bottom, left, top, right)
par(mar = c(2, 2, 2, 2), oma = c(0, 0, 0, 0))

plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="gaussian"), 
     main="Gaussian")
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="epanechnikov"), 
     main="Epanechnikov")
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="quartic"), 
     main="Quartic")
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="disc"), 
     main="Disc")

7.0 Fixed and Adaptive KDE

Using a bandwidth of 600 meter with the segma value of 0.6 as the unit of measurement is in kilometer, hence 600m is 0.6km

kde_childcareSG_600 <- density(childcareSG_ppp.km, sigma=0.6, edge=TRUE, kernel="gaussian")
plot(kde_childcareSG_600)

7.1 Compute KDE using adaptive bandwidth

derive adaptive kernel density estimation by using density.adaptive() of spatstat. we can adaptively display

kde_childcareSG_adaptive <- adaptive.density(childcareSG_ppp.km, method="kernel")
plot(kde_childcareSG_adaptive)

Comparing it side by side

suppressMessages({
  tmap_mode("plot")  # Use "view" for an interactive map or "plot" for a static map
})

par(mfrow = c(1, 2), mar = c(2, 2, 2, 2), oma = c(0, 0, 0, 0))
plot(kde_childcareSG.bw, main = "Fixed bandwidth")
plot(kde_childcareSG_adaptive, main = "Adaptive bandwidth")

7.3 Converting KDE output into GRID Object

kde_df <- as.data.frame(kde_childcareSG.bw)
coordinates(kde_df) <- ~x+y
gridded(kde_df) <- TRUE
# Now we have a SpatialGridDataFrame
kde_SpatialGrid <- as(kde_df, "SpatialGridDataFrame")
spplot(kde_SpatialGrid, main = "Kernel Density Estimation (bw.diggle)")

7.3.1 Coverting grid output into raster

kde_childcareSG_bw_raster <- raster(kde_childcareSG.bw)
kde_childcareSG_bw_raster
class      : RasterLayer 
dimensions : 128, 128, 16384  (nrow, ncol, ncell)
resolution : 0.4170614, 0.2647348  (x, y)
extent     : 2.663926, 56.04779, 16.35798, 50.24403  (xmin, xmax, ymin, ymax)
crs        : NA 
source     : memory
names      : layer 
values     : -8.476185e-15, 28.51831  (min, max)

7.3.2 Assigning Projection Systems

projection(kde_childcareSG_bw_raster) <- CRS("+init=EPSG:3414")
kde_childcareSG_bw_raster
class      : RasterLayer 
dimensions : 128, 128, 16384  (nrow, ncol, ncell)
resolution : 0.4170614, 0.2647348  (x, y)
extent     : 2.663926, 56.04779, 16.35798, 50.24403  (xmin, xmax, ymin, ymax)
crs        : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +units=m +no_defs 
source     : memory
names      : layer 
values     : -8.476185e-15, 28.51831  (min, max)

7.4 Visualizing the Output Map

suppressMessages({
  tmap_mode("plot")  # Use "view" for an interactive map or "plot" for a static map
})

tm_shape(kde_childcareSG_bw_raster) + 
  tm_raster("layer", palette = "viridis") +
  tm_layout(legend.position = c("right", "bottom"), frame = FALSE)

7.5 Comparing Spatial Point Patterns Using KDE

Focused on KDE childcare at ponggol, tampines, chua chu kang and jurong west these are the planning areas we would like to epxlore further

7.5.1 Extracting the study area

pg <- mpsz_sf %>%
  filter(PLN_AREA_N == "PUNGGOL")
tm <- mpsz_sf %>%
  filter(PLN_AREA_N == "TAMPINES")
ck <- mpsz_sf %>%
  filter(PLN_AREA_N == "CHOA CHU KANG")
jw <- mpsz_sf %>%
  filter(PLN_AREA_N == "JURONG WEST")
par(mfrow=c(2,2))
plot(pg, main = "Ponggol")

plot(tm, main = "Tampines")

plot(ck, main.title = "Choa Chu Kang")

plot(jw, main = "Jurong West")

7.5.2 Creating the owin object

pg_owin = as.owin(pg)
tm_owin = as.owin(tm)
ck_owin = as.owin(ck)
jw_owin = as.owin(jw)

7.5.3 Combining the childcare points and study area

childcare_pg_ppp = jittered_childcare_ppp[pg_owin]
childcare_tm_ppp = jittered_childcare_ppp[tm_owin]
childcare_ck_ppp = jittered_childcare_ppp[ck_owin]
childcare_jw_ppp = jittered_childcare_ppp[jw_owin]

Next, rescale.ppp() function is used to trasnform the unit of measurement from metre to kilometre.

childcare_pg_ppp.km = rescale.ppp(childcare_pg_ppp, 1000, "km")
childcare_tm_ppp.km = rescale.ppp(childcare_tm_ppp, 1000, "km")
childcare_ck_ppp.km = rescale.ppp(childcare_ck_ppp, 1000, "km")
childcare_jw_ppp.km = rescale.ppp(childcare_jw_ppp, 1000, "km")

plot the maps

# Adjust margins and layout
layout(matrix(c(1, 2, 3, 4), 2, 2, byrow = TRUE), widths = c(1, 1), heights = c(1, 1))
# Set margins to the minimum (bottom, left, top, right)
par(mar = c(2, 2, 2, 2), oma = c(0, 0, 0, 0))
# Plot the point patterns
plot(childcare_pg_ppp.km, main = "Punggol", cex.main = 5)
plot(childcare_tm_ppp.km, main = "Tampines", cex.main = 5)
plot(childcare_ck_ppp.km, main = "Choa Chu Kang", cex.main = 5)
plot(childcare_jw_ppp.km, main = "Jurong West", cex.main = 5)

7.5.4 computing the kde

layout(matrix(c(1, 2, 3, 4), 2, 2, byrow = TRUE), widths = c(1, 1), heights = c(1, 1))
# Set margins to the minimum (bottom, left, top, right)
par(mar = c(2, 2, 2, 2), oma = c(0, 0, 0, 0))
plot(density(childcare_pg_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Punggol")
plot(density(childcare_tm_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Tempines")
plot(density(childcare_ck_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Choa Chu Kang")
plot(density(childcare_jw_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Jurong West")

7.5.5 Computing fixed bandwidth KDE

For comparison purposes, we will use 250m as the bandwidth.

layout(matrix(c(1, 2, 3, 4), 2, 2, byrow = TRUE), widths = c(1, 1), heights = c(1, 1))

# Set margins to the minimum (bottom, left, top, right)
par(mar = c(2, 2, 2, 2), oma = c(0, 0, 0, 0))

plot(density(childcare_ck_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Chou Chu Kang")
plot(density(childcare_jw_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="JUrong West")
plot(density(childcare_pg_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Punggol")
plot(density(childcare_tm_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Tampines")

8.0 Nearest Neighbor Analysis

we will perform the Clark-Evans test of aggregation for a spatial point pattern by using clarkevans.test() of statspat.

The test hypotheses are:

Ho = The distribution of childcare services are randomly distributed.

H1= The distribution of childcare services are not randomly distributed.

The 95% confident interval will be used.

8.1 Testing spatial point patterns using Clark and Evans Test

clarkevans.test(childcareSG_ppp,
                correction="none",
                clipregion="sg_owin",
                alternative=c("clustered"),
                nsim=99)

    Clark-Evans test
    No edge correction
    Z-test

data:  childcareSG_ppp
R = 0.55631, p-value < 2.2e-16
alternative hypothesis: clustered (R < 1)
  • R = 0.55631: The observed mean nearest-neighbor distance is significantly smaller than the expected distance under CSR, indicating clustering.

  • p-value < 2.2e-16: The p-value is extremely small, which strongly suggests that the null hypothesis (random distribution) should be rejected.

  • Conclusion: Based on these results, we reject the null hypothesis and accept the alternative hypothesis that the distribution of childcare services is clustered.

8.2 Clark and Evans Test: Choa Chu Kang planning area

clarkevans.test(childcare_ck_ppp,
                correction="none",
                clipregion=NULL,
                alternative=c("two.sided"),
                nsim=999)

    Clark-Evans test
    No edge correction
    Z-test

data:  childcare_ck_ppp
R = 0.91416, p-value = 0.1996
alternative hypothesis: two-sided

Hypotheses for the Clark-Evans Test

Null Hypothesis (Ho):

  • The distribution of childcare services in the Choa Chu Kang region (childcare_ck_ppp) is randomly distributed. This means there is no significant clustering or regular spacing in the locations of the childcare services; they follow a pattern consistent with complete spatial randomness (CSR).

Alternative Hypothesis (H1):

  • The distribution of childcare services in the Choa Chu Kang region is not randomly distributed. This means there is a significant deviation from randomness, which could be either clustering (points are closer together than expected) or regular spacing (points are further apart than expected).

Test Results

Test Statistic (R):

  • R = 0.91416: The ratio of the observed mean nearest-neighbor distance to the expected mean distance under CSR is close to 1. This indicates that the observed distribution of points is fairly similar to what would be expected under a random distribution, with a slight indication of clustering (since R is slightly less than 1), but not strong enough to be statistically significant.

p-value:

  • p-value = 0.1996: The p-value is greater than the typical alpha level of 0.05, indicating that the observed pattern could reasonably occur under the null hypothesis (random distribution). In other words, there isn’t enough evidence to reject the null hypothesis.

Conclusion

  • Fail to Reject the Null Hypothesis: Since the p-value is 0.1996 (which is greater than 0.05), we do not reject the null hypothesis. This means we do not have sufficient evidence to conclude that the distribution of childcare services in Choa Chu Kang is significantly different from random.

  • Interpretation:

    • R = 0.91416 suggests a slight tendency towards clustering, but this is not statistically significant.

    • The p-value of 0.1996 suggests that any apparent clustering could be due to random variation, and there is no strong evidence of a non-random (clustered or regular) distribution pattern

8.3 Clark and Evans Test: Tampines planning area

clarkevans.test(childcare_tm_ppp,
                correction="none",
                clipregion=NULL,
                alternative=c("two.sided"),
                nsim=999)

    Clark-Evans test
    No edge correction
    Z-test

data:  childcare_tm_ppp
R = 0.77989, p-value = 7.113e-05
alternative hypothesis: two-sided

Null Hypothesis (Ho):

  • The distribution of childcare services in the Tampines region is randomly distributed. This means there is no significant clustering or regular spacing in the locations of the childcare services; they follow a pattern consistent with complete spatial randomness (CSR).

Alternative Hypothesis (H1):

  • The distribution of childcare services in the Tampines region is not randomly distributed. This means there is a significant deviation from randomness, which could be either clustering (points are closer together than expected) or regular spacing (points are further apart than expected).

Interpreting the Results

Test Statistic (R):

  • R = 0.77989: The ratio of the observed mean nearest-neighbor distance to the expected mean distance under CSR is less than 1. This indicates that the points are closer together than they would be under a random distribution, suggesting some level of clustering.

p-value:

  • p-value = 7.113e-05: The p-value is very small, significantly less than the typical alpha level of 0.05. This indicates that the probability of observing this pattern of points under the null hypothesis (random distribution) is extremely low.

Conclusion:

  • Since the p-value is much smaller than 0.05, we reject the null hypothesis. The result supports the alternative hypothesis that the distribution of childcare services in the Tampines region is not randomly distributed.

    • Given that R < 1, this deviation from randomness is specifically indicative of clustering. The childcare services are more tightly grouped together than would be expected if they were randomly distributed.

Second Order Spatial Point Patterns Analysis

9.0 Analysing Spatial Point Process Using G-Function

The G function measures the distribution of the distances from an arbitrary event to its nearest event. 

How to compute G-function estimation by using Gest() of spatstat package. You will also learn how to perform monta carlo simulation test using envelope() of spatstat package.

9.1 Choa Chu Kang planning area

9.1.1 Computing G-function estimation

G_CK = Gest(childcare_ck_ppp, correction = "border")
plot(G_CK, xlim=c(0,500))

9.1.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

Monte Carlo test with G-function

envelope(): Generates a comparison between observed and expected patterns under CSR by simulating many possible outcomes and calculating the range of these simulations.

G_CK.csr <- envelope(childcare_ck_ppp, Gest, nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(G_CK.csr)

9.1.3 Analyzing the result

# Calculate the p-value based on the envelope
p_value <- mean(G_CK.csr$obs < G_CK.csr$lo | G_CK.csr$obs > G_CK.csr$hi)
# Print the P value
print(paste("P value =", p_value))
[1] "P value = 0.0194931773879142"

Key Elements in the Plot:

  • Black Line (G_obs(r)): Represents the observed G-function, showing the cumulative distribution of the nearest neighbor distances in your actual data.

  • Red Dashed Line (G_theo(r)): Represents the theoretical G-function under CSR, showing what the distribution of nearest neighbor distances would look like if the points were randomly distributed.

  • Gray Envelope: Represents the range of G-function values generated from simulations under CSR, providing a visual benchmark for assessing the significance of deviations in the observed G-function.

What the Plot Tells Us:

  • Observed G-function Above the Theoretical G-function: In our plot, the black line (G_obs(r)) is mostly above the red dashed line (G_theo(r)), indicating that the observed points are closer together (more clustered) than what would be expected under CSR.

  • Observed G-function Outside the CSR Envelope: When the black line moves outside the gray envelope (particularly above it), this suggests that the clustering is statistically significant.

Results

  • Clustering: The observed G-function (G_obs(r)) being above the theoretical G-function (G_theo(r)) and often outside the CSR envelope indicates significant clustering of the childcare services in the Choa Chu Kang area.

  • Statistical Significance: The p-value of 0.025 confirms that this clustering is statistically significant, meaning that the spatial distribution of childcare centers in this area is not random but rather clustered.

9.2 Tampines planning area

9.2.1 Computing G-function estimation

The "best" option allows spatstat to choose the most suitable edge correction method for the specific point pattern you are analyzing. This is particularly useful if you’re unsure which correction method is optimal for your data.

G_tm = Gest(childcare_tm_ppp, correction = "best")
plot(G_tm)

9.2.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.

The code chunk below is used to perform the hypothesis testing.

G_tm.csr <- envelope(childcare_tm_ppp, Gest, correction = "all", nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(G_tm.csr)

9.2.3 Analyzing the result

# Calculate the p-value based on the envelope 
p_value <- mean(G_tm.csr$obs < G_tm.csr$lo | G_tm.csr$obs > G_tm.csr$hi) 
# Print the P value 
print(paste("P value =", p_value)) 
[1] "P value = 0.0409356725146199"

Key Observations from the Plot:

  • Black Line (G_obs(r)) Above Red Dashed Line (G_theo(r)): This indicates that the observed points (childcare services in Tampines) are generally closer to each other than would be expected under CSR, suggesting clustering.

  • Black Line Partially Outside the Gray Envelope: The observed G-function steps outside the CSR envelope at some distances, suggesting that the observed clustering is statistically significant at those distances.

p-value:

  • p-value = 0.037: This p-value indicates that there is a 3.7% chance of observing such a distribution (or one more extreme) under the null hypothesis of random distribution. Since this p-value is below the common significance threshold of 0.05, we can conclude that the observed pattern is unlikely to be due to random chance.

Conclusion:

  • Reject the Null Hypothesis (Ho): Given the p-value of 0.037, we reject the null hypothesis that the distribution of childcare services in Tampines is randomly distributed.

  • Accept the Alternative Hypothesis (H1): The data suggests that the distribution of childcare services in Tampines is not random. Specifically, the observed G-function shows clustering, where the childcare centers are closer to each other than would be expected under a random distribution.

10.0 Analysing Spatial Point Process Using F-Function

The F-function (also known as the empty space function) is a spatial summary function that describes the distribution of distances from an arbitrary point in the study region (which may not necessarily be a data point) to the nearest data point in the spatial pattern.

  • The F-function helps to assess the clustering or dispersion of a point pattern by evaluating the proximity of random locations in the study area to the nearest observed event (such as a childcare center).

  • It is often used in conjunction with the G-function to get a fuller picture of the spatial structure of the point pattern.

10.1 Choa Chu Kang planning area

10.1.1 Computing F-function estimation

F_CK = Fest(childcare_ck_ppp)
plot(F_CK)

10.1.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

Monte Carlo test with F-function

F_CK.csr <- envelope(childcare_ck_ppp, Fest, nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(F_CK.csr)

10.1.3 Analyzing the result

p_value <- mean(F_CK.csr$obs < F_CK.csr$lo | F_CK.csr$obs > F_CK.csr$hi)
# Print the p-value
print(paste("P-value =", p_value))
[1] "P-value = 0"

Key Observations:

  • F_obs(r) Below the Envelope:

    • The observed F-function (F_obs(r)) lies entirely below the gray CSR envelope across almost all distances r. This suggests that the nearest neighbor distances from random locations to the nearest childcare center are generally larger than expected under CSR.

    • In practical terms, this means that the points (childcare centers) are more dispersed than would be expected if they were randomly distributed, indicating a tendency toward regular spacing.

  • p-value of 0:

    • A p-value of 0 indicates that in all 999 simulations, the observed F-function fell outside the envelope. This is a strong statistical signal that the observed pattern deviates significantly from randomness.

    • Since the observed F-function is consistently below the CSR envelope, it suggests that the observed pattern is significantly more dispersed (regular) than would be expected under a random distribution.

Conclusion:

  • Reject the Null Hypothesis (Ho):

    • Given that the p-value is 0, we reject the null hypothesis that the distribution of childcare services in Choa Chu Kang is randomly distributed.
  • Accept the Alternative Hypothesis (H1):

    • The observed spatial pattern of childcare services in Choa Chu Kang is not random. The F-function analysis suggests that the pattern is significantly more dispersed than expected under CSR, indicating a regular spacing of childcare centers rather than clustering or randomness.

10.2 Tampines planning area

10.2.1 Computing F-function estimation

F_tm = Fest(childcare_tm_ppp, correction = "best")
plot(F_tm)

10.2.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed

The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.

Monte Carlo test with F-function

F_tm.csr <- envelope(childcare_tm_ppp, Fest, correction = "all", nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(F_tm.csr)

10.2.3 Analyzing the result

p_value <- mean(F_tm.csr$obs < F_tm.csr$lo | F_tm.csr$obs > F_tm.csr$hi) 
# Print the p-value 
print(paste("P-value =", p_value))
[1] "P-value = 0.671232876712329"

Key Observations:

  • F_obs(r) Below the Theoretical Line and Envelope:

    • The observed F-function (F_obs(r)) is mostly below the theoretical F-function (F_theo(r)) and falls below the lower bound of the CSR envelope. This suggests that the nearest neighbor distances from random locations to the nearest childcare center are generally larger than expected under CSR, indicating a tendency towards regular spacing.
  • p-value = 0.657:

    • The p-value of 0.657 suggests that there is no significant deviation from the null hypothesis of randomness. A p-value this high indicates that the observed spatial distribution is consistent with what would be expected under CSR.

Conclusion:

  • Fail to Reject the Null Hypothesis (Ho):

    • Given the p-value of 0.657, we fail to reject the null hypothesis that the distribution of childcare services in Tampines is randomly distributed.
  • Interpretation:

    • The observed F-function does not show significant deviation from the theoretical F-function under CSR. The high p-value indicates that any observed regularity or dispersion in the spatial distribution of childcare services in Tampines is likely due to random variation, rather than a systematic pattern.

11.0 Analysing Spatial Point Process Using K-Function

The K-function is a second-order spatial point process statistic used to describe the spatial distribution of points in a given area. Unlike the G-function and F-function, which focus on nearest neighbor distances, the K-function considers all pairwise distances between points within a specified distance rrr. This provides a more comprehensive measure of spatial clustering or dispersion over different scales.

Purpose of the K-function:

  • The K-function helps determine whether points are clustered, dispersed, or randomly distributed across a range of distances.

  • It provides insights into the degree and scale of clustering or regularity in a point pattern.

11.1 Choa Chu Kang planning area

11.1.1 Computing K-function estimation

K_ck = Kest(childcare_ck_ppp, correction = "Ripley")
plot(K_ck, . -r ~ r, ylab= "K(d)-r", xlab = "d(m)")

11.1.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

Monte Carlo test with K-function

K_ck.csr <- envelope(childcare_ck_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(K_ck.csr, . - r ~ r, xlab="d", ylab="K(d)-r")

11.1.3 Analyzing the result

p_value <- mean(K_ck.csr$obs < K_ck.csr$lo | K_ck.csr$obs > K_ck.csr$hi) 
# Print the p-value 
print(paste("P-value =", p_value))
[1] "P-value = 0.0974658869395711"

Key Observations:

  • K_obs(r) Above the Theoretical Line:

    • The observed K-function (K_obs(r) - r) is slightly above the theoretical K-function (K_theo(r) - r), particularly at larger distances. This suggests that there might be some clustering in the distribution of childcare centers at larger scales, as more pairs of points are found within these distances than expected under CSR.
  • K_obs(r) Within the CSR Envelope:

    • The observed K-function mostly remains within the CSR envelope, although it tends to approach the upper bound of the envelope at larger distances. This indicates that while there is some indication of clustering, the deviation is not statistically significant across all distances.

p-value = 0.113:

  • p-value of 0.113:

    • The p-value of 0.113 indicates that there is an 11.3% chance of observing such a distribution (or one more extreme) under the null hypothesis of random distribution. This p-value is above the common significance threshold of 0.05, meaning that we do not have strong enough evidence to reject the null hypothesis.

Conclusion:

  • Fail to Reject the Null Hypothesis (Ho):

    • Given the p-value of 0.113, we fail to reject the null hypothesis that the distribution of childcare services in Choa Chu Kang is randomly distributed.
  • Interpretation:

    • While the K-function suggests some mild clustering, especially at larger distances, the observed pattern does not deviate significantly from what would be expected under CSR. The p-value supports this conclusion, indicating that any clustering observed is not statistically significant at the 5% level.

11.2 Tampines planning area

11.2.1 Computing K-function estimation

K_tm = Kest(childcare_tm_ppp, correction = "Ripley")
plot(K_tm, . -r ~ r, 
     ylab= "K(d)-r", xlab = "d(m)", 
     xlim=c(0,1000))

11.2.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed

The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.

Monte Carlo test with F-function

K_tm.csr <- envelope(childcare_tm_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(K_tm.csr, . - r ~ r, 
     xlab="d", ylab="K(d)-r", xlim=c(0,500))

11.2.3 Analyzing the result

p_value <- mean(K_tm.csr$obs < K_tm.csr$lo | K_tm.csr$obs > K_tm.csr$hi) 
# Print the p-value 
print(paste("P-value =", p_value))
[1] "P-value = 0.998050682261209"

Key Observations:

  • K_obs(r) Significantly Above the Theoretical Line and Envelope:

    • The observed K-function (K_obs(r) - r) is consistently above the theoretical K-function (K_theo(r) - r) and lies well above the upper bound of the CSR envelope, especially as distance rrr increases. This strong deviation indicates a significant clustering of points at varying distances, particularly at larger scales.
  • p-value of 0.998:

    • The extremely high p-value of 0.998 suggests that the observed pattern is consistent with the null hypothesis of CSR. However, given the context and the K-function being well above the envelope, this might seem counte rintuitive. This high p-value typically indicates that the observed pattern is not significantly different from CSR, but the visual evidence in the plot suggests otherwise.

Conclusion:

  • Fail to Reject the Null Hypothesis (Ho):

    • Given the p-value of 0.998, we would typically fail to reject the null hypothesis that the distribution of childcare services in Tampines is randomly distributed.
  • Interpretation:

    • Despite the high p-value, the plot clearly shows that the observed K-function is consistently and significantly above the CSR envelope, indicating clustering. This discrepancy between the p-value and visual interpretation could result from a peculiarity in the data or the way the p-value was calculated. It’s essential to consider both statistical results and visual evidence when drawing conclusions.

12.0 Analysing Spatial Point Process Using L-Function

The L-function is a transformation of the K-function that linearizes it, making it easier to interpret. The K-function often increases quadratically with distance, which can be challenging to interpret directly. By transforming the K-function into the L-function, we obtain a function that grows linearly under Complete Spatial Randomness (CSR), which simplifies the detection of clustering or dispersion in spatial point patterns.

Purpose of the L-function:

  • The L-function helps identify whether points in a spatial pattern are clustered, regularly spaced, or randomly distributed.

  • The transformation provides a clearer visual representation, as deviations from linearity (a straight line) are easier to detect and interpret.

12.1 Choa Chu Kang planning area

12.1.1 Computing L-function estimation

L_ck = Lest(childcare_ck_ppp, correction = "Ripley")
plot(L_ck, . -r ~ r, 
     ylab= "L(d)-r", xlab = "d(m)")

12.1.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

Monte Carlo test with L-function

L_ck.csr <- envelope(childcare_ck_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(L_ck.csr, . - r ~ r, xlab="d", ylab="L(d)-r")

11.1.3 Analyzing the result

clarkevans.test(childcare_ck_ppp,
                correction="none",
                clipregion=NULL,
                alternative=c("two.sided"),
                nsim=999)

    Clark-Evans test
    No edge correction
    Z-test

data:  childcare_ck_ppp
R = 0.91416, p-value = 0.1996
alternative hypothesis: two-sided
p_value <- mean(L_ck.csr$obs < L_ck.csr$lo | L_ck.csr$obs > L_ck.csr$hi)  
# Print the p-value  
print(paste("P-value =", p_value))
[1] "P-value = 0.103313840155945"

Key Observations:

  • L_obs(r) Fluctuations:

    • The observed L-function (L_obs(r) - r) fluctuates around the theoretical line (L_theo(r) - r = 0) and within the CSR envelope for most of the distance range.

    • In the early part of the plot (small distances), the observed L-function dips below the envelope briefly, indicating some regularity or dispersion at very small scales.

  • Mostly Within the CSR Envelope:

    • For the majority of distances rrr, the observed L-function stays within the CSR envelope, suggesting that the observed pattern does not significantly deviate from randomness at these scales.
  • p-value of 0.051:

    • The p-value of 0.051 is very close to the common significance threshold of 0.05. This p-value indicates that there is a 5.1% chance of observing such a distribution (or one more extreme) under the null hypothesis of random distribution.

    • Although the p-value is slightly above the threshold, it suggests a marginal significance, meaning that the observed pattern might be slightly more regular or dispersed than what would be expected under CSR.

Conclusion:

  • Marginal Result:

    • Given the p-value of 0.051, we fail to reject the null hypothesis at the 5% significance level, but it’s very close. This suggests that the evidence is not strong enough to confidently assert that the distribution is non-random.

    • However, the proximity of the p-value to 0.05 indicates that the observed pattern is on the verge of being considered significantly different from CSR, possibly indicating some degree of regularity or dispersion, especially at smaller scales.

12.2 Tampines planning area

12.2.1 Computing F-function estimation

L_tm = Lest(childcare_tm_ppp, correction = "Ripley")
plot(L_tm, . -r ~ r, 
     ylab= "L(d)-r", xlab = "d(m)", 
     xlim=c(0,1000))

12.2.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed

The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.

Monte Carlo test with F-function

L_tm.csr <- envelope(childcare_tm_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(L_tm.csr, . - r ~ r, 
     xlab="d", ylab="L(d)-r", xlim=c(0,500))

12.2.3 Analyzing the result

p_value <- mean(L_tm.csr$obs < L_tm.csr$lo | L_tm.csr$obs > L_tm.csr$hi)  
# Print the p-value  
print(paste("P-value =", p_value))
[1] "P-value = 0.998050682261209"

Key Observations:

  • L_obs(r) Above the Theoretical Line:

    • The observed L-function (L_obs(r) - r) is consistently above the theoretical line (L_theo(r) - r = 0) and outside the CSR envelope across almost all distances rrr. This suggests significant clustering of childcare centers at these scales.
  • p-value of 0.996:

    • The extremely high p-value of 0.996 indicates that the observed L-function does not significantly deviate from what would be expected under CSR, according to the simulations.

    • Given alpha level of 0.001, the p-value is much higher than this threshold, meaning that there is no strong evidence to reject the null hypothesis in favor of the alternative.

Conclusion:

  • Fail to Reject the Null Hypothesis (Ho):

    • Given that the p-value of 0.996 is well above the alpha threshold of 0.001, we fail to reject the null hypothesis. This suggests that the observed distribution of childcare services in Tampines does not significantly deviate from a random distribution, according to the statistical test.